operation
According to the dictionary Webster’s 1913, which can be accessed through
\htmladdnormallinkHyperDictionary.comhttp://www.hyperdictionary.com/, mathematical
meaning of the word operation is: “some transformation
to be
made upon quantities”. Thus, operation is similar
to mapping or function. The most
general mathematical definition of operation can be made as follows:
Definition 1
Operation # defined on the sets X1,X2,…,Xn with values in X is a mapping from Cartesian product X1×X2×⋯×Xn to X, i.e.
#:X1×X2×⋯×Xn⟶X. |
Result of operation is usually denoted by one of the following notation:
-
•
x1#x2#⋯#xn
-
•
#(x1,…,xn)
-
•
(x1,…,xn)#
The following examples show variety of the concept operation used in mathematics.
Examples
-
1.
Arithmetic operations: addition
(http://planetmath.org/Addition), subtraction
, multiplication (http://planetmath.org/Multiplication), division. Their generalization
leads to the so-called binary operations
, which is a basic concept for such algebraic structures
as groups and rings.
-
2.
Operations on vectors in the plane (ℝ2).
-
–
Multiplication by a scalar. Generalization leads to vector spaces
.
-
–
Scalar product
. Generalization leads to Hilbert spaces.
-
–
-
3.
Operations on vectors in the space (ℝ3).
-
–
Cross product
. Can be generalized for the vector space of arbitrary finite dimension
, see vector product in general vector spaces.
-
–
Triple product.
-
–
-
4.
Some operations on functions.
-
–
Composition.
-
–
Function inverse
.
-
–
In the case when some of the sets Xi are equal to the values set X, it is usually said that operation is defined just on X. For such operations, it could be interesting to consider their action on some subset U⊂X. In particular, if operation on elements from U always gives an element from U, it is said that U is closed under this operation. Formally it is expressed in the following definition.
Definition 2
Let operation #:X1×X2×⋯×Xn⟶X is defined on X, i.e. there exists k≥1 and indexes 1≤j1<j2<⋯<jk≤n such that Xj1=Xj2=⋯=Xjk=X. For simplicity, let us assume that ji=i. A subset U⊂X is said to be closed under operation # if for all u1,u2,…,uk from U and for all xj∈Xjj>k holds:
#(u1,u2,…,uk,xk+1,xk+2,…,xn)∈U. |
The next examples illustrates this definition.
Examples
-
1.
Vector space V over a field K is a set, on which the following two operations are defined:
-
–
multiplication by a scalar:
⋅:K×V⟶V -
–
addition
+:V×V⟶V.
Of course these operations need to satisfy some properties (for details see the entry vector space). A subset W⊂V, which is closed under these operations, is called vector subspace.
-
–
-
2.
Consider collection
of all subsets of the real numbers ℝ, which we denote by 2ℝ. On this collection, binary operation intersection of sets is defined:
∩:2ℝ×2ℝ⟶2ℝ. Collection of sets ℭ⊂2ℝ:
ℭ:= is closed under this operation.
Title | operation |
Canonical name | Operation |
Date of creation | 2013-03-22 14:57:23 |
Last modified on | 2013-03-22 14:57:23 |
Owner | rspuzio (6075) |
Last modified by | rspuzio (6075) |
Numerical id | 10 |
Author | rspuzio (6075) |
Entry type | Definition |
Classification | msc 03E20 |
Related topic | Function |
Related topic | Mapping |
Related topic | Transformation |
Related topic | BinaryOperation |
Defines | closed under |
Defines | arithmetic operation |